Ztransform

The Z-transform is a mathematical tool used to analyze linear time-invariant discrete-time systems and signals. For a discrete-time sequence x[n], the bilateral Z-transform is defined as X(z) = sum_{n=-∞}^{∞} x[n] z^{-n}, where z is a complex variable. A unilateral Z-transform often used for causal systems is X(z) = sum_{n=0}^{∞} x[n] z^{-n}. The Z-transform maps a time-domain sequence into a complex function, facilitating manipulation such as multiplication by polynomials and rational function representations.

The region of convergence (ROC) is the set of all z for which the Z-transform sum converges. The ROC depends on the signal: right-sided sequences have ROC outside a outermost pole, left-sided sequences have ROC inside an innermost pole, and two-sided sequences have ROC between poles. The Z-transform is typically a rational function, with poles (where X(z) → ∞) and zeros (where X(z) = 0) determined by the sequence.

Poles and zeros provide insight into system behavior. For finite- or ARMA-like sequences, X(z) can be expressed as a ratio of polynomials. Stability in discrete time is achieved when the ROC includes the unit circle |z| = 1, ensuring bounded input produces bounded output (BIBO stability). The location of poles near the unit circle indicates slow decay or potential instability.

Key properties include linearity, time shifting (Z{x[n−n0]} = z^{−n0} X(z)), and the convolution theorem (Z{x[n] ∗ h[n]} = X(z) H(z)). For causal sequences, the initial value can be retrieved by x[0] = lim_{z→∞} X(z). The Z-transform is closely related to the Laplace transform via z = e^{sT}, linking discrete-time analysis to continuous-time theory.

Applications span digital filter analysis and design, stability assessment, spectral analysis, and the solution of difference equations.